Critical frequency of Faraday Waves?

[Anjish]

TL;DR Summary: A fluid on a vertically vibrating plate will, upon reaching a certain frequency and acceleration, produce standing waves on its surface. These are called Faraday waves, first described by Michael Faraday in 1831. Faraday waves are still an active area of research today, more than 150 years after their initial discovery. In current research the terms “Faraday instability” and “standing gravity waves” are also used for this phenomenon.

Please help me understand the relation of the critical frequency with the resonant frequency of Faraday waves to exist in a container filled with liquid like water. How can we calculate it mathematically?

IF I want to investigate it by changing the viscosity, depth of the liquid and temperature and how it affects the amplitude and critical frequency of the Faraday waves, if there is no mathematical equation, can we deduce one? can there be a relationship? and also can anyone suggest more relations or investigations I can take with faraday waves?

I am little confused with existing literature and the physics of faraday waves, especially the mathematics of it if it exists. Please help me. This is my current research question
"How do variations in the viscosity of the liquid (or different liquids) and changes in the temperature of the vessel affect the wavelength, amplitude, and frequency of Faraday waves?"

Is it valid? IF yes can you help me with the methodology part and physics and mathematics part of it? what is the maths I can use or formulas to calculate? Is water propagation and dispersion relevant? Can this science be used- https://en.wikipedia.org/wiki/Dispe...e table below, the,the phase and group speeds.

 

[berkeman]

Welcome to PF.

This is my current research question
"How do variations in the viscosity of the liquid (or different liquids) and changes in the temperature of the vessel affect the wavelength, amplitude, and frequency of Faraday waves?"

What year are you in university? Is this project for a thesis?

 

[Anjish]

No, I am in grade 12th doing IB Diploma program and this is my investigation for my Extended Essay(EE) which is 4000 word research paper. Can you please guide me on how I can go with this topic and investigation? I am facing difficulty with comprehending research papers and mostly their mathematics. So, Guide on the mathematics and investigation would be of immense help to me. Thank you

 

[berkeman]

Since this is for your schoolwork, I will move this thread to the schoolwork forums for you now.

Also, on schoolwork projects like this, we require that you show your own work first before we can offer tutorial help to you. Also, asking us to read through your multiple PDF references and describe the subject to you is a pretty big ask.

How did you pick this project? What parts of the math to you understand so far? How much calculus have you studied so far in school?

(I will send you a separate message with hints for using LaTeX to post math equations here)

 

[Anjish]

I understand every branch of mathematics except in-depth calculus, I can solve differential equations but don't know integration and all. But I am studying that now so I can use it for my investigation, also I attached those papers as I thought it would be helpful for someone to go through maybe one and give me advice.
I tried to read the papers, I understood the theory but cannot understand the critical frequency aspect of it, shall I do the trial and error method to get one critical frequency, or is there any way to find it mathematically and then check it experimentally? Attached here is one equation used in the change in viscosity in amplitude, and it show this equation , can you explain me or share some resources to understand the maths?

 

[berkeman]

Can this science be used- https://en.wikipedia.org/wiki/Dispe...e table below, the,the phase and group speeds.

I'm not sure how that Wikipedia article is relevant.

Instead, have a look at this one to see if it helps: https://en.wikipedia.org/wiki/Faraday_wave

 

[Anjish]

Can you review by current investigation question? how shall I approach it?